Phase Crystals.

Superconducting order owes its properties to broken symmetry of the U (1) phase. In the presence of competing orders, or in pairbreaking environments, usual superconductivity may give way to modified condensates. Here we describe one such alternative, where the low-temperature ground state is more ordered due to formation of a periodic modulation of the phase. This state is manifested by spatially-periodic superflow patterns and circulating currents which break time-reversal symmetry. We demonstrate using microscopic theory how it can be realized near edges of superconductors that host flat bands of zero-energy Andreev bound states. More generally, we trace the origin of phase crystallization to non-local properties of the gradient energy, which implies existence of similar pattern-forming instabilities in many other systems.